Intrinsic brackets and the $L_{\infty }$-deformation theory of bialgebras

Abstract

We show that there exists a Lie a bracket on the cohomology of any type of (bi)algebras over an operad or a PROP, induced by a strongly homotopy Lie structure on the defining cochain complex, such that the associated "quantum" master equation captures deformations. This in particular implies the existence of a Lie bracket on the Gerstenhaber-Schack cohomology of a bialgebra that extends the classical intrinsic bracket on the Hochschild cohomology, giving an affirmative answer to an old question about the existence of such a bracket.

The final version. To appear in Journal of Homotopy and Related Structures

References in corpus (3)

• Lie theory for nilpotent L-infinity algebras
• Infinitesimal bialgebras, pre-Lie and dendriform algebras
• Operads up to Homotopy and Deformations of Operad Maps

Cited by in corpus (4)

• Deformation theory of representations of prop(erad)s
• Koszul duality for operadic categories
• The L_\infty-deformation complex of diagrams of algebras
• A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids